The Yee scheme, also called the finite difference time domain (FDTD) method, developed by Kane Yee in the 1960's is one of the most popular numerical methods for the approximation of Maxwell's equations in the time domain. It simultaneously discretizes the electric and magnetic fields by staggering these fields in space and time leading to an explicit method that has second order temporal and spatial accuracy. A number of different extensions of the Yee scheme to dispersive media (frequency dependent material) were developed in the 1990s. One of these extensions appends to Maxwell's equations a system of ordinary differential equations that govern the dynamic evolution of the (macroscopic) polarization excited by the propagating electric field. For accuracy, the discretization of such dispersive Maxwell models require resolving scales that are a tenth or a thousandth of the incident electromagnetic pulse. In addition the schemes are more dispersive than the standard Yee scheme.
To reduce the effects of numerical dispersion, we consider spatial higher order extensions of the Yee scheme for dispersive media, while maintaining the second order accuracy in time. In this talk we will consider such staggered (2,2M) schemes for Maxwell dispersive models which are second order accurate in time and 2M order accurate in space, for arbitrary integer M>0. We will discuss the derivation of closed form analytical stability criteria for the staggered (2,2M) FDTD methods. In addition we also derive numerical dispersion relations for these schemes which provides information about the expected accuracy of the method. The key result required to perform the stability and dispersion analyses for arbitrary M > 0 is the equivalence of the symbol of the 2M order finite difference approximation of the first order derivative operator ∂/∂z with the truncation of an appropriate series expansion of the symbol of ∂/∂z.